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Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients
Journal of Mathematical Analysis and Applications 262, 554–563 (2001) doi:10.1006/jmaa.2001.7554, available online at http://www.idealibrary.com on
Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients K. Farahmand and A. Grigorash1 Department of Mathematics, University of Ulster at Jordanstown, County Antrim, BT37 0QB, United Kingdom E-mails: [email protected]; [email protected] Submitted by William F. Ames Received October 11, 1999
In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0 + η1 cos θ + η2 cos 2θ + · · · + ηn cos nθ in 0 2π, where the coefficients ηj = aj + ιbj , and aj nj=1 and bj nj=1 are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known. 2001 Academic Press Key Words: Number of complex zeros; real roots; complex roots; random trigonometric polynomials; random algebraic polynomials; Jacobian of transformation; Adler’s theorem; coordinate transform; density of zeros.
1. INTRODUCTION Let aj nj=1 and bj nj=1 be sequences of normally distributed random variables with mean 0 and variance 1, and ηj = aj + ιbj . There are many known results for the expected number of reals of a random algebraic polynomial of the form Pn x ≡ Px =
n j=1
ηj xj
These results include the pioneering works of Dunnage [4] and Sambandham [8], and the more recent works of Wilkins [10], which are 1
Work supported by a scholarship from the Vice-Chancellor of the University of Ulster. 554
0022-247X/01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
complex zeros of trigonometric polynomials
555
reviewed in a comprehensive book by Bharucha-Reid and Sambandham [3]. Also there are some results concerning the density of the complex roots of Px. For the latest developments in this direction, see [5] and [6]. However, the method used for Px is not directly applicable when we consider other types of polynomials, such as the trigonometric case. In this paper, we consider the random trigonometric polynomials Tn z ≡ T z =
n j=1
ηj cos jz
(1.1)
where z ∈ z is a complex variable, z = ϕ + ιψ ϕ ψ ∈ . Whereas the period of cos jz is a real number 2π, we can suppose 0 ≤ ϕ < 2π −∞ < ψ < ∞. For the random vectors a1 an T and b1 bn T composed from the real and imaginary parts of the polynomial coefficients, we introduce notation a and b correspondingly. Let 1 = T z and 2 = T z denote the real and imaginary parts of T z, respectively. Then = 1 2 T is a two-dimensional random field. Denote ∂1 /∂r ∂2 /∂r ∇ = ∂1 /∂θ ∂2 /∂θ a matrix of (random) coordinate transformation r θ → 1 2 , where r and θ stand for the polar representation of the complex variable z, with z = reιθ . Let S be a compact subset in the complex z plane z such that on the boundary ∂S of S there are no points for which = 0 T z = 0, and S does not contain any points satisfying both = 0 and ∇ = 0. Also assume S does not include the real axis in the plane z , that is, r θ θ = 0 ⊂ S. Suppose νzn S to be the number of points z ∈ S for which T z = 0, and Eνzn S its expected value. Then from Adler’s theorem [1, Theorem 5.1.1, p. 95] (see also Ibragimov and Zeitouni [7] or Shepp and Vanderbei [9]), we have Eνzn S = hnz r θ dr dθ S
and
hnz r θ = E det ∇ = 0 p0 0
(1.2)
where px1 x2 is a two-dimensional joint density function of the random vector . Now if we consider a transformation z → w with w = eιz
(1.3)
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and denote ρ = e−ψ , we obtain the polar representation of w, with w = ρeιϕ w ∈ w . From the transformation (1.3) we obtain polynomial (1.1) in the form Tn w =
1 2
n j=1
ηj wj + w−j
(1.4)
Let X1 = Tn w and X2 = Tn w. Then X = X1 X2 T is a two-dimensional random field in the w plane w . Denote ∂X1 /∂r ∂X2 /∂r ∇X = ∂X1 /∂θ ∂X2 /∂θ Obviously, by applying the transformation (1.3) to the set S in the z plane z , we obtain a set in the w plane w . Denote this set as . We prove the following theorem. Theorem 1. With the preceding assumptions and also if does not contain any points satisfying both X = 0 and ∇X = 0, then Eνzn S = re−r sin θ hnw e−r sin θ r cos θ dr dθ (1.5) S
where
n n 1 hnw ρ ϕ = j 2 ρ2j − eι2jϕ ρ2j + eι2jϕ πρ j=−n j=−n j=0 j=0 2 2 2 n n n 1 − jρ2j + jeι2jϕ ρ2j + eι2jϕ j=−n πρ j=−n j=−n j=0 j=0 j=0
(1.6)
2. PROOF OF THE THEOREM The main idea of the proof is to apply Adler’s theorem [1, Theorem 5.1.1, p. 95] and its corollary [1, p. 97] to polynomial (1.4). To this end, we establish that satisfies the assumptions of this corollary stated in the foregoing text. Because (1.3) is a continuous one-to-one transform and S is a compact, then is a compact. Also, because Tn z = =
n j=1 1 2
ηj cos jz =
n j=1
1 2
n j=1
ηj eιjz + e−ιjz
ηj wj + w−j = Tn w
(2.1)
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we can see that a zero of Tn z is transformed to a zero of Tn w. By definition of a border and continuity of the transformation (1.3), the boundary ∂S of the set S is transformed to the boundary ∂ of the set . Using the fact that (1.3) is a one-to-one transform that maps zeros of Tn z to zeros of Tn w, we conclude that whereas S does not contain infinitely many zeros of Tn w, the set may contain only a finite number of zeros of Tn w. Remark 1. In the previous section, we excluded real axis ψ = 0 from S. Consequently, the central unity circle ρ = 1 is excluded from . Remark 2. For the sake of convenience in the calculations that follow, it is useful to exclude the point w = 0 from consideration. This is justified by the fact that the coordinate origin w = 0 in the complex w plane w corresponds to the point z = ∞ in the z plane z . z = ϕ2 + ψ2 Arg z = arctanψ/ϕ The point z = ∞ was excluded from consideration because the interval −∞ ∞ containing values of ψ is open. Let νwn be the number of points w ∈ for which Tn w = 0. Then Eνwn =
hnw ρ ϕ dρ dϕ
hnw ρ ϕ = E det ∇X X = 0pρϕ 0 0
(2.2) (2.3)
where pρϕ x1 x2 is a two-dimensional joint density function of the random vector X. Because the zeros of T z are mapped to the zeros of T w and is the image of S, νzn S = νwn and
Eνzn S = Eνwn
(2.4)
Now whereas the Jacobian of the coordinate transformation ρ ϕ −→ r θ is re−r sin θ , from (2.2) and (2.4) we can obtain Eνzn S =
S
re−r sin θ hnw e−r sin θ r cos θ dr dθ
Now to determine the component hnw , we need the following lemma. The proof of the theorem is an immediate consequence of the lemma.
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Lemma 1. Provided all the conditions imposed on and X are satisfied, then for all n, the density function hnw ρ ϕ is n n 1 ρ2j + eι2jϕ j 2 ρ2j − eι2jϕ hnw ρ ϕ = πρ j=−n j=−n j=0
j=0
2 2 n n n 1 − jρ2j + jeι2jϕ ρ2j + eι2jϕ πρ j=−n j=−n j=−n j=0 j=0 j=0
2
3. DENSITY OF COMPLEX ZEROS Polynomial (1.4) can be rewritten as Tn w =
1 2
n j=−n j=0
ηj wj
From the definitions of X1 and X2 it follows that X1 =
1 2
X2 =
1 2
and
n j=−n j=0
n j=−n j=0
ρj aj cos jϕ − bj sin jϕ ρj aj sin jϕ + bj cos jϕ
Whereas 2 n 1 jρj aj sin jϕ + bj cos jϕ det ∇X = 4ρ j=−n +
j=0
n j=−n j=0
2 jρ aj cos jϕ − bj sin jϕ j
det ∇X ≥ 0. Then the absolute value sign that appears in the formula for hnw can be eliminated and, therefore, hnw can be rewritten as hnw ρ ϕ = Edet ∇XX = 0pρϕ 0 0
(3.1)
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559
Expanding det ∇X and reducing similar terms at products of random coefficients aj and bj , we obtain the following representation of det ∇X: det ∇X =
n 1 jkρj+k aj ak + bj bk cosj − kϕ 4ρ j k=−n j k=0
We define
+ 2aj bk sinj − kϕ
(3.2)
!ab = E a − Eab − EbT
a generalized covariance matrix of two vectors a and b, and !ab X = !ab − !aX !−1 XX !Xb Based on the assumption that all the scalar random variables involved are independent and normally distributed, from standard methods in multivariate analysis (see, for example, Anderson [2]), we obtain !aa X !ab X cova bX = 0 = !ba X !bb X and T EaX = 0 = Ea − !aX !−1 XX EX
Whereas the distribution of aj and bj is central, Ea = 0 and EX = 0. This implies EaX = 0 = 0. Analogously, EbX = 0 = 0. To calculate cova bX = 0, we first show that !−1 XX exists. To this end, we rewrite X1 and X2 in the form X1 =
1 2
X2 =
1 2
n
(3.3)
aj ρj − ρ−j sin jϕ + bj ρj + ρ−j cos jϕ
(3.4)
j=1
aj ρj + ρ−j cos jϕ − bj ρj − ρ−j sin jϕ
and n j=1
Let det !XX = V 2 , and whereas aj and bj are independent, we obtain 1 0 1 !−1 XX = V 0 1 where, from (3.3) and (3.4), V =
1 4
n j=1
ρ2j + ρ−2j + 2 cos 2jϕ =
1 4
n j=−n j=0
ρ2j + cos 2jϕ
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Remark 3. To show that V is strictly positive, we note that 2 cos 2jϕ ≥ −2 and ρ2j + ρ−2j ≥ 2. Therefore, V is nonnegative. Also, because we can rewrite V in the form of V =
1 4
n
ρ2j + ρ−2j + 2 cos2 jϕ − 2
j=1
we conclude that the condition V = 0 is equivalent to n
n ρ2j + ρ−2j − 2 + 2 cos2 jϕ = 0
j=1
j=1
For this equality to hold, it is necessary that nj=1 2 cos2 jϕ = 0, which implies cos jϕ = 0 for every j. This is obviously impossible. Thus, V is strictly positive. Now we obtain !aa X !bb X !ab X . To this end, given that !aa = !bb = I and !ab = 0, simple algebra leads us to !aa X = !bb X = I −
1 j+k + ρ−j−k cosj − kϕ ρ 4V
+ ρj−k + ρk−j cosj + kϕ
n×n
and !ab X = −
1 j+k − ρ−j−k sinj − kϕ ρ 4V
+ ρj−k − ρk−j sinj + kϕ n×n
Therefore, we can easily derive the expressions for Eaj ak + bj bk X = 0 and Eaj bk X = 0, and after all the necessary simplifications, we have Edet ∇XX = 0 =
n n 1 1 ρ2j + ρ−2j − 2 cos 2jϕ − jk ρ2j+k + ρ−2j+k 2ρ j=1 8ρV j k=−n j k=0
+ 2 cos 2j − kϕ − 2 cos 2j + kϕ 2 2 n n n ι2jϕ 2j 1 2 2j 1 ι2jϕ = j ρ −e jρ + je (3.5) − 2ρ j=−n 8ρV j=−n j=−n
− ρ2j−k + ρ
j=0
2k−j
j=0
j=0
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561
When w is fixed, X1 and X2 are random variables distributed according to normal (Gaussian) law. Their joint density function is −1 n 2 pρ ϕ x1 x2 = ρ2j + cos 2jϕ π j=−n j=0
−1 n ρ2j + cos 2jϕ × exp x21 + x22 j=−n π 2
j=0
which gives −1 n 2 ρ2j + cos 2jϕ pρ ϕ 0 0 = π j=−n j=0
This together with (3.5) proves the lemma.
4. ASYMPTOTICS It is of particular interest to study the density function hnw when n tends to infinity. Compared with other types of polynomials, such as random algebraic polynomials, for our type of random polynomials, this asymptotic value of hnw is surprisingly simple: lim hnw ρ ϕ =
n→∞
ρ πρ2 − 12
To derive this relationship, we first rewrite hnw in the form involving trigonometric rather than exponential functions: −1 n 1 ρ2j + cos 2jϕ hnw ρ ϕ = πρ j=−n j=0
×
n
j=−n
2
j ρ2j − cos 2jϕ −
n
−1
ρ2j + cos 2jϕ
j=−n j=0
j=0
2 2j × jρ + j cos 2jϕ + j sin 2jϕ j=−n j=−n j=−n
n j=0
2
n j=0
2
n j=0
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Decomposing hnw to the sum of three items hnw ρ ϕ = where n
j=−n j=0
I1 = n
j 2 ρ2j − cos 2jϕ
j=−n j=0
ρ2j + cos 2jϕ n
j=−n j=0
I2 = n and
n
I2 = n
j=−n j=0
j=−n j=0
− I2 − I3 ,
2
jρ2j
ρ2j + cos 2jϕ
j=−n j=0
1 I πρ 1
2
j sin 2jϕ
ρ2j + cos 2jϕ
we reduce the problem of calculating limn→∞ hnw to calculation of the limits of these three items. Using the formulae for the n well known n n n sums ρ2j , jρ2j , j 2 ρ2j , j=−n j = 0 j=−n j = 0 j=−n j = 0 j=−n j=0 cos 2jϕ, n n 2 j sin 2jϕ, and j cos 2jϕ, one can easily show that j=−n j=0 j=−n j=0
lim I1 =
n→∞
limn→∞ n2 −
2 ρ2 −1
limn→∞ n +
ρ2 +1 ρ2 −12
when ρ > 1,
limn→∞ n2 +
2ρ2 ρ2 −1
limn→∞ n +
ρ2 ρ2 +1 ρ2 −12
when ρ < 1,
limn→∞ n2 +
2ρ2 ρ2 −1
limn→∞ n +
ρ4 ρ2 −12
when ρ < 1,
limn→∞ n2 −
2 ρ2 −1
limn→∞ n +
1 ρ2 −12
when ρ > 1.
and lim I2 =
n→∞
The foregoing expressions for limn→∞ I1 and limn→∞ I2 clearly show that lim I1 − I2 =
n→∞
ρ2 ρ2 − 12
for every ρ. Noting that limn→∞ I3 = 0, we therefore conclude that lim hnw ρ ϕ =
n→∞
ρ2 1 πρ ρ2 − 12
and the formula to demonstrate follows.
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REFERENCES 1. R. J. Adler, “The Geometry of Random Fields,” Wiley, Chichester, 1981. 2. T. W. Anderson, “Introduction to Multivariate Statistical Analysis,” Wiley, New York, 1984. 3. A. T. Bharucha-Reid and M. Sambandham, “Random Polynomials,” Academic Press, New York, 1986. 4. J. E. A. Dunnage, The number of real zeros of a class of random algebraic polynomials, Proc. London Math. Soc. 18 (1968), 439–460. 5. K. Farahmand, Complex roots of a random algebraic polynomial, J. Math. Anal. Appl. 210 (1997), 724–730. 6. K. Farahmand, “Topics in Random Polynomials,” Addison–Wesley Longman, London, 1998. 7. I. A. Ibragimov and O. Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), 2427–2441. 8. M. Sambandham, On the average number of real zeros of a class of random algebraic curves, Pacific J. Math. 81 (1979), 207–215. 9. L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), 4365–4383. 10. J. E. Wilkins, An asymptotic expansion for the expected number of real zeros of a random polynomial, Proc. Amer. Math. Soc. 103 (1988), 1249–1258.
Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients K. Farahmand and A. Grigorash1 Department of Mathematics, University of Ulster at Jordanstown, County Antrim, BT37 0QB, United Kingdom E-mails: [email protected]; [email protected] Submitted by William F. Ames Received October 11, 1999
In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0 + η1 cos θ + η2 cos 2θ + · · · + ηn cos nθ in 0 2π, where the coefficients ηj = aj + ιbj , and aj nj=1 and bj nj=1 are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known. 2001 Academic Press Key Words: Number of complex zeros; real roots; complex roots; random trigonometric polynomials; random algebraic polynomials; Jacobian of transformation; Adler’s theorem; coordinate transform; density of zeros.
1. INTRODUCTION Let aj nj=1 and bj nj=1 be sequences of normally distributed random variables with mean 0 and variance 1, and ηj = aj + ιbj . There are many known results for the expected number of reals of a random algebraic polynomial of the form Pn x ≡ Px =
n j=1
ηj xj
These results include the pioneering works of Dunnage [4] and Sambandham [8], and the more recent works of Wilkins [10], which are 1
Work supported by a scholarship from the Vice-Chancellor of the University of Ulster. 554
0022-247X/01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
complex zeros of trigonometric polynomials
555
reviewed in a comprehensive book by Bharucha-Reid and Sambandham [3]. Also there are some results concerning the density of the complex roots of Px. For the latest developments in this direction, see [5] and [6]. However, the method used for Px is not directly applicable when we consider other types of polynomials, such as the trigonometric case. In this paper, we consider the random trigonometric polynomials Tn z ≡ T z =
n j=1
ηj cos jz
(1.1)
where z ∈ z is a complex variable, z = ϕ + ιψ ϕ ψ ∈ . Whereas the period of cos jz is a real number 2π, we can suppose 0 ≤ ϕ < 2π −∞ < ψ < ∞. For the random vectors a1 an T and b1 bn T composed from the real and imaginary parts of the polynomial coefficients, we introduce notation a and b correspondingly. Let 1 = T z and 2 = T z denote the real and imaginary parts of T z, respectively. Then = 1 2 T is a two-dimensional random field. Denote ∂1 /∂r ∂2 /∂r ∇ = ∂1 /∂θ ∂2 /∂θ a matrix of (random) coordinate transformation r θ → 1 2 , where r and θ stand for the polar representation of the complex variable z, with z = reιθ . Let S be a compact subset in the complex z plane z such that on the boundary ∂S of S there are no points for which = 0 T z = 0, and S does not contain any points satisfying both = 0 and ∇ = 0. Also assume S does not include the real axis in the plane z , that is, r θ θ = 0 ⊂ S. Suppose νzn S to be the number of points z ∈ S for which T z = 0, and Eνzn S its expected value. Then from Adler’s theorem [1, Theorem 5.1.1, p. 95] (see also Ibragimov and Zeitouni [7] or Shepp and Vanderbei [9]), we have Eνzn S = hnz r θ dr dθ S
and
hnz r θ = E det ∇ = 0 p0 0
(1.2)
where px1 x2 is a two-dimensional joint density function of the random vector . Now if we consider a transformation z → w with w = eιz
(1.3)
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and denote ρ = e−ψ , we obtain the polar representation of w, with w = ρeιϕ w ∈ w . From the transformation (1.3) we obtain polynomial (1.1) in the form Tn w =
1 2
n j=1
ηj wj + w−j
(1.4)
Let X1 = Tn w and X2 = Tn w. Then X = X1 X2 T is a two-dimensional random field in the w plane w . Denote ∂X1 /∂r ∂X2 /∂r ∇X = ∂X1 /∂θ ∂X2 /∂θ Obviously, by applying the transformation (1.3) to the set S in the z plane z , we obtain a set in the w plane w . Denote this set as . We prove the following theorem. Theorem 1. With the preceding assumptions and also if does not contain any points satisfying both X = 0 and ∇X = 0, then Eνzn S = re−r sin θ hnw e−r sin θ r cos θ dr dθ (1.5) S
where
n n 1 hnw ρ ϕ = j 2 ρ2j − eι2jϕ ρ2j + eι2jϕ πρ j=−n j=−n j=0 j=0 2 2 2 n n n 1 − jρ2j + jeι2jϕ ρ2j + eι2jϕ j=−n πρ j=−n j=−n j=0 j=0 j=0
(1.6)
2. PROOF OF THE THEOREM The main idea of the proof is to apply Adler’s theorem [1, Theorem 5.1.1, p. 95] and its corollary [1, p. 97] to polynomial (1.4). To this end, we establish that satisfies the assumptions of this corollary stated in the foregoing text. Because (1.3) is a continuous one-to-one transform and S is a compact, then is a compact. Also, because Tn z = =
n j=1 1 2
ηj cos jz =
n j=1
1 2
n j=1
ηj eιjz + e−ιjz
ηj wj + w−j = Tn w
(2.1)
complex zeros of trigonometric polynomials
557
we can see that a zero of Tn z is transformed to a zero of Tn w. By definition of a border and continuity of the transformation (1.3), the boundary ∂S of the set S is transformed to the boundary ∂ of the set . Using the fact that (1.3) is a one-to-one transform that maps zeros of Tn z to zeros of Tn w, we conclude that whereas S does not contain infinitely many zeros of Tn w, the set may contain only a finite number of zeros of Tn w. Remark 1. In the previous section, we excluded real axis ψ = 0 from S. Consequently, the central unity circle ρ = 1 is excluded from . Remark 2. For the sake of convenience in the calculations that follow, it is useful to exclude the point w = 0 from consideration. This is justified by the fact that the coordinate origin w = 0 in the complex w plane w corresponds to the point z = ∞ in the z plane z . z = ϕ2 + ψ2 Arg z = arctanψ/ϕ The point z = ∞ was excluded from consideration because the interval −∞ ∞ containing values of ψ is open. Let νwn be the number of points w ∈ for which Tn w = 0. Then Eνwn =
hnw ρ ϕ dρ dϕ
hnw ρ ϕ = E det ∇X X = 0pρϕ 0 0
(2.2) (2.3)
where pρϕ x1 x2 is a two-dimensional joint density function of the random vector X. Because the zeros of T z are mapped to the zeros of T w and is the image of S, νzn S = νwn and
Eνzn S = Eνwn
(2.4)
Now whereas the Jacobian of the coordinate transformation ρ ϕ −→ r θ is re−r sin θ , from (2.2) and (2.4) we can obtain Eνzn S =
S
re−r sin θ hnw e−r sin θ r cos θ dr dθ
Now to determine the component hnw , we need the following lemma. The proof of the theorem is an immediate consequence of the lemma.
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Lemma 1. Provided all the conditions imposed on and X are satisfied, then for all n, the density function hnw ρ ϕ is n n 1 ρ2j + eι2jϕ j 2 ρ2j − eι2jϕ hnw ρ ϕ = πρ j=−n j=−n j=0
j=0
2 2 n n n 1 − jρ2j + jeι2jϕ ρ2j + eι2jϕ πρ j=−n j=−n j=−n j=0 j=0 j=0
2
3. DENSITY OF COMPLEX ZEROS Polynomial (1.4) can be rewritten as Tn w =
1 2
n j=−n j=0
ηj wj
From the definitions of X1 and X2 it follows that X1 =
1 2
X2 =
1 2
and
n j=−n j=0
n j=−n j=0
ρj aj cos jϕ − bj sin jϕ ρj aj sin jϕ + bj cos jϕ
Whereas 2 n 1 jρj aj sin jϕ + bj cos jϕ det ∇X = 4ρ j=−n +
j=0
n j=−n j=0
2 jρ aj cos jϕ − bj sin jϕ j
det ∇X ≥ 0. Then the absolute value sign that appears in the formula for hnw can be eliminated and, therefore, hnw can be rewritten as hnw ρ ϕ = Edet ∇XX = 0pρϕ 0 0
(3.1)
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559
Expanding det ∇X and reducing similar terms at products of random coefficients aj and bj , we obtain the following representation of det ∇X: det ∇X =
n 1 jkρj+k aj ak + bj bk cosj − kϕ 4ρ j k=−n j k=0
We define
+ 2aj bk sinj − kϕ
(3.2)
!ab = E a − Eab − EbT
a generalized covariance matrix of two vectors a and b, and !ab X = !ab − !aX !−1 XX !Xb Based on the assumption that all the scalar random variables involved are independent and normally distributed, from standard methods in multivariate analysis (see, for example, Anderson [2]), we obtain !aa X !ab X cova bX = 0 = !ba X !bb X and T EaX = 0 = Ea − !aX !−1 XX EX
Whereas the distribution of aj and bj is central, Ea = 0 and EX = 0. This implies EaX = 0 = 0. Analogously, EbX = 0 = 0. To calculate cova bX = 0, we first show that !−1 XX exists. To this end, we rewrite X1 and X2 in the form X1 =
1 2
X2 =
1 2
n
(3.3)
aj ρj − ρ−j sin jϕ + bj ρj + ρ−j cos jϕ
(3.4)
j=1
aj ρj + ρ−j cos jϕ − bj ρj − ρ−j sin jϕ
and n j=1
Let det !XX = V 2 , and whereas aj and bj are independent, we obtain 1 0 1 !−1 XX = V 0 1 where, from (3.3) and (3.4), V =
1 4
n j=1
ρ2j + ρ−2j + 2 cos 2jϕ =
1 4
n j=−n j=0
ρ2j + cos 2jϕ
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Remark 3. To show that V is strictly positive, we note that 2 cos 2jϕ ≥ −2 and ρ2j + ρ−2j ≥ 2. Therefore, V is nonnegative. Also, because we can rewrite V in the form of V =
1 4
n
ρ2j + ρ−2j + 2 cos2 jϕ − 2
j=1
we conclude that the condition V = 0 is equivalent to n
n ρ2j + ρ−2j − 2 + 2 cos2 jϕ = 0
j=1
j=1
For this equality to hold, it is necessary that nj=1 2 cos2 jϕ = 0, which implies cos jϕ = 0 for every j. This is obviously impossible. Thus, V is strictly positive. Now we obtain !aa X !bb X !ab X . To this end, given that !aa = !bb = I and !ab = 0, simple algebra leads us to !aa X = !bb X = I −
1 j+k + ρ−j−k cosj − kϕ ρ 4V
+ ρj−k + ρk−j cosj + kϕ
n×n
and !ab X = −
1 j+k − ρ−j−k sinj − kϕ ρ 4V
+ ρj−k − ρk−j sinj + kϕ n×n
Therefore, we can easily derive the expressions for Eaj ak + bj bk X = 0 and Eaj bk X = 0, and after all the necessary simplifications, we have Edet ∇XX = 0 =
n n 1 1 ρ2j + ρ−2j − 2 cos 2jϕ − jk ρ2j+k + ρ−2j+k 2ρ j=1 8ρV j k=−n j k=0
+ 2 cos 2j − kϕ − 2 cos 2j + kϕ 2 2 n n n ι2jϕ 2j 1 2 2j 1 ι2jϕ = j ρ −e jρ + je (3.5) − 2ρ j=−n 8ρV j=−n j=−n
− ρ2j−k + ρ
j=0
2k−j
j=0
j=0
complex zeros of trigonometric polynomials
561
When w is fixed, X1 and X2 are random variables distributed according to normal (Gaussian) law. Their joint density function is −1 n 2 pρ ϕ x1 x2 = ρ2j + cos 2jϕ π j=−n j=0
−1 n ρ2j + cos 2jϕ × exp x21 + x22 j=−n π 2
j=0
which gives −1 n 2 ρ2j + cos 2jϕ pρ ϕ 0 0 = π j=−n j=0
This together with (3.5) proves the lemma.
4. ASYMPTOTICS It is of particular interest to study the density function hnw when n tends to infinity. Compared with other types of polynomials, such as random algebraic polynomials, for our type of random polynomials, this asymptotic value of hnw is surprisingly simple: lim hnw ρ ϕ =
n→∞
ρ πρ2 − 12
To derive this relationship, we first rewrite hnw in the form involving trigonometric rather than exponential functions: −1 n 1 ρ2j + cos 2jϕ hnw ρ ϕ = πρ j=−n j=0
×
n
j=−n
2
j ρ2j − cos 2jϕ −
n
−1
ρ2j + cos 2jϕ
j=−n j=0
j=0
2 2j × jρ + j cos 2jϕ + j sin 2jϕ j=−n j=−n j=−n
n j=0
2
n j=0
2
n j=0
562
farahmand and grigorash
Decomposing hnw to the sum of three items hnw ρ ϕ = where n
j=−n j=0
I1 = n
j 2 ρ2j − cos 2jϕ
j=−n j=0
ρ2j + cos 2jϕ n
j=−n j=0
I2 = n and
n
I2 = n
j=−n j=0
j=−n j=0
− I2 − I3 ,
2
jρ2j
ρ2j + cos 2jϕ
j=−n j=0
1 I πρ 1
2
j sin 2jϕ
ρ2j + cos 2jϕ
we reduce the problem of calculating limn→∞ hnw to calculation of the limits of these three items. Using the formulae for the n well known n n n sums ρ2j , jρ2j , j 2 ρ2j , j=−n j = 0 j=−n j = 0 j=−n j = 0 j=−n j=0 cos 2jϕ, n n 2 j sin 2jϕ, and j cos 2jϕ, one can easily show that j=−n j=0 j=−n j=0
lim I1 =
n→∞
limn→∞ n2 −
2 ρ2 −1
limn→∞ n +
ρ2 +1 ρ2 −12
when ρ > 1,
limn→∞ n2 +
2ρ2 ρ2 −1
limn→∞ n +
ρ2 ρ2 +1 ρ2 −12
when ρ < 1,
limn→∞ n2 +
2ρ2 ρ2 −1
limn→∞ n +
ρ4 ρ2 −12
when ρ < 1,
limn→∞ n2 −
2 ρ2 −1
limn→∞ n +
1 ρ2 −12
when ρ > 1.
and lim I2 =
n→∞
The foregoing expressions for limn→∞ I1 and limn→∞ I2 clearly show that lim I1 − I2 =
n→∞
ρ2 ρ2 − 12
for every ρ. Noting that limn→∞ I3 = 0, we therefore conclude that lim hnw ρ ϕ =
n→∞
ρ2 1 πρ ρ2 − 12
and the formula to demonstrate follows.
complex zeros of trigonometric polynomials
563
REFERENCES 1. R. J. Adler, “The Geometry of Random Fields,” Wiley, Chichester, 1981. 2. T. W. Anderson, “Introduction to Multivariate Statistical Analysis,” Wiley, New York, 1984. 3. A. T. Bharucha-Reid and M. Sambandham, “Random Polynomials,” Academic Press, New York, 1986. 4. J. E. A. Dunnage, The number of real zeros of a class of random algebraic polynomials, Proc. London Math. Soc. 18 (1968), 439–460. 5. K. Farahmand, Complex roots of a random algebraic polynomial, J. Math. Anal. Appl. 210 (1997), 724–730. 6. K. Farahmand, “Topics in Random Polynomials,” Addison–Wesley Longman, London, 1998. 7. I. A. Ibragimov and O. Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), 2427–2441. 8. M. Sambandham, On the average number of real zeros of a class of random algebraic curves, Pacific J. Math. 81 (1979), 207–215. 9. L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), 4365–4383. 10. J. E. Wilkins, An asymptotic expansion for the expected number of real zeros of a random polynomial, Proc. Amer. Math. Soc. 103 (1988), 1249–1258.